The systems and methods herein disclosed pertain to the general problem of calculating the steady-state powerflow of DC electrical power networks. In the context of power electronics and vehicular DC power systems, this is also referred to as calculating the operating point of the system. More specifically, the methods described here address the computation of the powerflow in the presence of constant power loads, which is a basic problem in power transmission and distribution systems, as well as in the nonlinear analysis of DC circuits.
The powerflow problem, also known as loadflow, is the cornerstone calculation for the analysis of electrical power systems. It essentially consists in the calculation of voltages at all buses, given a set of prescribed constant-power injections (load or generation), and one or more swing buses which act as the voltage reference and provide ‘slack’ power. Powerflow studies are widely used for utility AC grids, but the problem is less commonly known as such in DC systems. Part of the reason is that utility-size meshed DC systems are still at their early stage of development. Another reason is that practitioners of DC power electronics do not refer to this problem by the same name, but instead just use the generic term of “nonlinear analysis” of the steady state (also operating point, bias point). Nevertheless, the underlying problem in DC is completely analogous to its counterpart in AC. In this regard, a clarification is in order: the problem known as “DC Loadflow” actually has very little to do with the DC powerflow discussed here. The “DC Loadflow” is a linearized approximation to the AC powerflow, a technique that is commonly used for a range of studies in AC utility grids, mostly for expansion planning and market-related studies.
In the context of onboard DC power distribution systems in spacecraft and the so-called more-electric aircraft, ships, and vehicles, the power network is commonly referred to as the Power Management and Distribution (PMAD) system. In modern PMAD systems, constant-power loads play a major role. These are mainly DC motors, as well as other tightly regulated loads. Constant-power loads also play important roles in smaller DC power electronic systems, such as computers. In all these cases, the powerflow analysis is performed in the framework of the standard nonlinear analysis for finding the DC operating point (or points) of the circuit. Classical tools such as the well-known program SPICE used for analog electrics design contain algorithms for this calculation.
Just like their counterparts in AC systems, the state of the art powerflow algorithms for DC systems have all one thing in common, which is their reliance on numerical iteration as the root-finding technique lying at the core of the procedure. Examples of these core techniques are Gauss-Seidel iteration and the widely used Newton-Raphson method, in its many variants. Some powerflow algorithms improve on the performance and convergence properties of the core iterative methods by using homotopy (continuation methods). Other algorithms utilize alternative techniques such as evolutionary algorithms or interval arithmetic, with the aim of improving the chance of global convergence (i.e. increasing the chance of converging to a solution regardless of the initial guess point), but they also make use of Newton-Raphson to zero-in on the solutions. A recent overview of the state of the art in this domain is provided in L. Trajkovic, “Exploration of the theory of DC Operating Points for Analog Circuit Design,” CASEO Workshop, IEEE International Symposium on Circuits and Systems (ISCAS), Beijing, China, 2013.
By contrast, U.S. Pat. Nos. 7,519,506 and 7,979,239 to A. Trias introduced a new AC powerflow method that is constructive and does not rely on contractive fixed-point iteration. The method, named the Holomorphic Embedding Loadflow Method (HELM), is based on a complex-valued embedding technique specifically tailored to exploit the particular algebraic nonlinearities of the powerflow problem (in contrast with numerical iteration, which is a generic root-finding technique that applies to any type of nonlinearity). It is important to remark that even though homotopy methods also use the concept of embedding, they have nothing in common with this new method. HELM uses techniques from complex analysis and algebraic geometry, and in particular draws heavily from the theory of complex algebraic curves. Its main advantages are that it unequivocally arrives to the operational solution if it exists, and conversely it unequivocally signals unfeasibility when the operational solution does not exist. The operational solution is the main solution of interest in power transmission and delivery systems; it is characterized by having high voltages (and low shunt currents) at buses everywhere, while all the other possible solutions have lower voltages (and higher shunt currents) at one or more buses. More precisely, HELM defines the operational solution as the one that continuously connects, in the analytic continuation sense, to a reference state at s=0 where all buses are energized, interconnected by the transmission network, but all connections to ground are open-circuited. The present disclosure extends the HELM method to DC systems, including not only constant power loads but also certain types of nonlinear devices.